A neutron initially at rest, decays into a proton, an electron, and an antineutrino. The ejected electron has a momentum of 1.4 10⁻²⁶ kg-m/s while the antineutrino has 6.4 10⁻²⁷ kg-m/s . Determine the recoil speed of the proton if the electron and the antineutrino are ejected along the same direction. The mass of the proton is 1.67 10⁻²⁷ kg .

Step-by-step solution

By the principle of conservation of linear momentum, the initial momentum of the neutron is equal to the vector sum of the final momenta of the decay products. p _p + p _e + p _ = 0 Since the electron and the antineutrino are ejected along the same direction, their momenta add up algebraically. The magnitude of the proton's momentum is equal to the sum of their momenta: p_p = p_e + p_ p_p = 1.4 10⁻²⁶ + 6.4 10⁻²⁷ p_p = 14 10⁻²⁷ + 6.4 10⁻²⁷ = 20.4 10⁻²⁷ kg-m/s The recoil speed of the proton is given by v_p = p_p m_p